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AIAA 2002-1008

Vortical Dynamics and Acoustic Response in Gas-Turbine Swirl-Stabilized Injectors Shanwu Wang, Shih-Yang Hsieh, Vigor Yang The Pennsylvania State University University Park, PA 16802, USA

40th Aerospace Sciences Meeting & Exhibit

14-17 January 2002 / Reno, NV

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AIAA 2002-1008

Vortical Dynamics and Acoustic Response in Gas-Turbine Swirl-Stabilized Injectors

Shanwu Wang∗, Shih-Yang Hsieh†, Vigor Yang‡

The Pennsylvania State University University Park, PA 16802, USA

Abstract

The vortical flow dynamics and acoustic response of a gas-turbine swirl-stabilized injector are investigated by means of a large-eddy-simulation (LES) technique. The flow passes the injector through three radial swirlers, which are counter-rotated with each other. The formulation includes Favre-averaged mass, momentum, and energy conservation equations in three dimensions. Several instability modes with well-defined frequencies, such as vortex breakdown and Kelvin-Helmholtz instabilities as well as their interactions, are observed in the flowfields. The method of proper orthogonal decomposition (POD) and spectral analysis is employed to identify the complex flow structures and several dominant frequencies in various regions. The acoustic response of the injector to imposed oscillations at the inlet is also studied over a wide range of frequency. Results show that external forcing has minor effects on the mean flow properties due to the broadband characteristic of turbulence. However, the instantaneous mass flux distribution at the injector outlet depends on the frequency of external excitation although the vortex breakdown is not sensitive to the imposed excitation in the current study. The radius of the central toroidal recirculation zone (CTRZ) varies during externally excitation and is out of phase with the overall mass flow rate at the injector outlet.

Introduction

Unsteady fluid dynamic interactions with chemical reactions in combustion chambers, a phenomenon commonly known as combustion instability, have hindered the development of gas turbine engines for years. The resultant flow oscillations may significantly shorten the lifetimes of a combustor and the associated turbomachinery components. Although it is well known that combustion dynamics results from thermoacoustic coupling (Rayleigh, 1945), the detailed mechanisms that lead to the onset and sustenance of combustion instability still are unclear and remain as a major technique problem, which must be solved.

In a combustion chamber, injectors play an important role in defining the stoichiometry and fluid

mechanics of the primary combustion zone. The effects of vorticity evolution in injectors appear in two areas. First, the vorticity evolution is a highly unsteady phenomenon that can induce pressure oscillation in the flowfield. Second, the strong shear layer associated with vorticity evolution may affect the breakup of liquid sheet from the injectors and the second breakup of liquid droplets and lumps (Lasheras and Hopfinger, 2000). These subsequently affect the fuel distribution and further affect the stoichiometry of the combustible mixtures. Therefore, the vorticity evolution in injector must be studied carefully.

Swirl-stabilized injectors have been commonly used in modern gas turbine engines as an aid to stabilize the high intensity combustion process and to promote efficient clean combustion. One of the most important flow characteristics produced by swirl-stabilized injectors is the central toroidal recirculation zone (CTRZ) (Gupta, 1984), which serves as a flame stabilization mechanism. Flows in this region are generally associated with high shear rates and turbulent intensity resulting from vortex breakdown. Such kind of flow oscillations may couple resonantly with acoustic modes in the combustor and subsequently lead to combustion instabilities. Since the flowfield generated by injector plays an important

∗ Ph.D. Student, Department of Mechanical

Engineering. † Research Associate, Department of Mechanical

Engineering, currently at GE Aircraft Engines. ‡ Professor, Department of Mechanical Engineering,

Fellow AIAA.

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role in defining the stoichiometry and fluid dynamics of the primary combustion zone, it is necessary to study the acoustic response of the injector for diagnosing the cause of combustion instability.

The studies of large flow structures in nonswirling flows are extensive, but limited effort was devoted to swirling flows, especially in complex geometries with strong turbulence (Sarpkaya, 1995). In recent years, computational models have become more useful in providing fundamental understanding of the physics involved. However, due to the limitations of computer resources, most studies utilized the Reynolds Averaged Navier Stocks (RANS) equations or two-dimensional LES to study the flowfields (Spall et al, 2000; Cannon et al, 2000; Guo et al. 2001). The purpose of this work is to conduct a comprehensive numerical analysis of the detailed flow structures in a swirl-stabilized injector over a broad range of operating conditions. The responses of the injector to various externally imposed disturbances are studied in detail.

Theoretical Formulation and Numerical Model

The present analysis is based on a large-eddy-simulation (LES) technique, in which large-scale turbulent structures are directly computed and small dissipative structures are modeled. Mathematically, the LES methodology begins with filtering of small-scale effects from large-scale motions in the full conservation equations.

The Favre-filtered conservation equations of mass, momentum, and energy can be expressed in the following conservative form:

0~

=∂

∂+

∂∂

j

j

xu

tρρ (1)

( )

j

sgsijij

j

ijjii

xxpuu

tu

∂−∂

=∂+∂

+∂∂ ττδρρ )~~(~

(2)

)~(

]~)~[(~

sgsjjiji

j

j

jtt

Hqux

xupe

te

−+∂∂

=∂

+∂+

∂∂

τ

ρρ

sgsijτ

(3)

where ρ, ui, p, et, qj and τij represent the density, velocity components, pressure, specific total energy, heat flux, and viscous stress tensor, respectively. The subgrid-scale terms in Eqs. (1)-(3), i.e., the subgrid stress and subgrid energy fluxes H , are closed by implementing an improved Smagorinsky model proposed by Erlebacher et al (1992). Details

of the filtered equations and the subgrid closure employed have been reported in the cited papers and therefore not presented here for brevity.

sgsj

The theoretical formulation outlined above is solved numerically by means of a density-based, finite-volume methodology. Spatial discretization is achieved using fourth- and second-order central difference schemes for the convective and viscous terms, respectively, in generalized coordinates (Rai et al, 1993). Temporal discretization is obtained using the two-step Adam-Bashforth predictor-corrector scheme. The message passing interface (MPI) parallel computing architecture with a multi-block domain composition technique is implemented to obtain further efficiency.

The method of characteristics (MOC) is used to treat the boundary conditions. The mass flux ( ),total temperature (Tm 0), axial velocity (ux), and angle between the radial (ur) and azimuthal velocity (uθ) are specified. The pressure is determined using a simplified one-dimensional momentum equation in the radial direction.

Results and Discussion

1. Computational Case Description

The injector under consideration consists of a mixing duct and a fuel nozzle located coaxially upstream of the mixing duct, as shown in Fig. 1. A detailed description is given in Graves (1997). The mixing duct includes a center cylindrical duct, two annular ducts, and three passages corresponding to these ducts. Those passages are spaced radially outward from each other. Three sets of radial air swirlers, denoted as S1, S2, and S3, respectively, are located upstream of the air-passages. The first and second swirlers are counter-rotating relative to the longitudinal axis. The radial swirler angles in the current study are 45°, -60°, and 70°, respectively.

A three-dimensional grid system is generated by rotating a two-dimensional grid system around the longitudinal axis. The external region downstream of the injector is also considered to provide a complete description of the flow development. The length and diameter of the computational domain are 10 times and 5 times of the injector diameter, respectively. The computational domain is carefully chosen such that the outer boundaries in the axial and radial directions are sufficiently far from the injector exit to minimize the propagation of boundary-induced disturbances into the injector. The baseline flow condition in the current study includes an ambient

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pressure of 1 atm, an inlet temperature of 293 K, and an inlet mass flow rate of 0.077 kg/s.

S1 S2 S3

S1 S2 S3

16 m

m

Air

Air

Swirler Fu

el

Noz

zle

Fig. 1 Schematic diagram of swirl-stabilized

injector. The selection of grid size in the injector region is based on the turbulence kinetic energy spectrum. The Reynolds number in the current study, based on the diameter and the bulk velocity at the injector outlet, is 2×105. The average gird size in the injector interior falls in the inertial sub-range of the turbulent kinetic energy spectrum. The entire grid system has 1.9 million points, of which 0.9 million points are located within the injector. A total number of 54 computational blocks are used for parallel processing

2. Unsteady Flow Evolution without External Forcing

Investigation was first conducted into the injector dynamics under conditions without externally imposed forcing. Figure 2 shows the vorticity magnitude contours on the x-y and y-z planes. Figure 3 shows an iso-surface of the azimuthal velocity. The flow patterns exhibit two features as follows. First is the vortex breakdown due to the radial-entry swirling flow. A CTRZ is found downstream of the centerbody due to this vortex breakdown. Because of the strong shear layer between the inlet passage and CTRZ, a strong vorticity layer is produced, which subsequently rolls, tilts, stretches, and breaks up into small bulbs. These vorticity bulbs are then convected downstream and interact with the surrounding flow structures. The strong shear layer helps break up the liquid steam injected from the fuel injection ports. The power spectral density (PSD) of the pressure oscillation in the CTRZ, as shown later, indicates the existence of broadband fluctuation with a dominant frequency around 4000 Hz. The second salient feature is the organized vortex shedding from the trailing edge of the guide vane between the first and second flow

passages. Because of the opposition of the swirlers vane angles, two highly counter-rotating flows at different velocities, merging at the rim tip, produce a strong shear layer that promotes mixing process. The vortices are generated and shed downstream sequentially. The vorticity magnitude contours in the A-A section of Fig. 2 also shows the existence of the Kelvin-Helmholtz instabilities in both the azimuthal and axial directions due to the large velocity difference across the guide vane. The ensuing influence on fuel/air mixing may be significant because the strong vortical flow interacts with the thin fuel film on the surface of the guide vane between the second and third passages.

100 154 236 363 559 859 1321 2031 3123 4802 7384 11353 17456 26840 41267 63450 97558 150000Ω: 1/S

A-A

A

A

Fig. 2 Snapshots of vorticity magnitude contours

Calculation was also conducted for case with low swirl number. Figure 4 shows a typical result for the inlet radial swirler vane angles of S1 = 30°, S2 = -40°, and S3 = 50°. The flow structures in term of vortex breakdown, vortex shedding and helical structure appear to be stronger than those with high swirl number cases. In Figure 4, a bubble type vortex breakdown is clearly observed in the central region and the vortex shedding near the guide vane is well organized.

The contours of the mean axial velocity and turbulence kinetic energy (tke) are presented in Figs. 5 and 6, respectively. Figure 5 clearly shows the existence of CTRZ, where the axial velocity is negative. High tke is observed in the downstream of the centerbody, where the vortex breakdown occurs, and the guide vane between the second and third passages, where the vortex shedding owing to the Kelvin-Helmholtz instability is dominant. The strong vortical motions in these regions promote the mixing between the fuel and air.

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Fig. 3 Iso-surface of azimuthal velocity (high

swirl number, S1= 45°, S2= -60°, S3= 70°)

Fig. 4 Iso-surface of azimuthal velocity (low swirl

number, S1= 30°, S2= -40°, S3= 50°)

-30 -26 -22 -18 -14 -10 -6 -2 2 6 10 14 18 22 26 30 34 38 42 46 50 54 58 62 66 70 74 78 82 86 90

• without forcing • 1500 Hz Forcing

ux:m/s

U

Fig. 5 Axial velocity contours of mean flowfields,

spatially averaged in the azimuthal direction

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200

• without forcing • 1500 Hz Forcing

tke: m2/s2

P02:02

P10:03

P14:10

Fig. 6 Contours of turbulence kinetic energy,

spatially averaged in the azimuthal direction.

The PSD is obtained to study the characteristics of the flowfield in the spectral space. Figure 7 shows the PSD of the pressure fluctuation at three different locations under conditions with and without forcing. The spectrum in high swirl-number case is broadband and no dominant frequency is observed as opposed to the lower swirl number flow study (Wang, et al., 2001). Different dominant frequencies are observed at different probes. A spectrum range around 1500Hz is dominant at probe 10:03 located at the Kelvin-Helmholtz instability dominant zone. This mode may correspond to the interaction between the Kelvin-Helmholtz instabilities and vortex breakdown. The POD result shown later indicates that the vortex shedding due to Kelvin-Helmholtz instabilities is at the frequency of 14000Hz. In the present work, due to the geometry constraints, the two instabilities, i.e., vortex breakdown and vortex shedding, will

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interact/compete with each other and the enhanced evolution of vortex breakdown suppresses the development of vortex shedding. At probe 14:10, which is inside of the CTRZ, a 4000Hz dominant frequency is clearly observed. The above results prove that complex characteristic behavior of the flowfield in the swirler-stabilized injector.

0 2000 4000 6000

101

102

103

fF=500 Hz

500Hz

frequency, Hzp'

,Pa

0 2000 4000 6000

101

102

103

fF=1500 Hz

1500Hz

frequency, Hz0 2000 4000 6000

101

102

103

fF=4000 Hz

4000Hz

p',P

a

0 2000 4000 6000

101

102

103

Free Oscillation

4000Hz

0 2000 4000 6000100

101

102

103

fF=500 Hz

500Hz

frequency, Hz

p',P

a

0 2000 4000 6000100

101

102

103

fF=1500 Hz

1500Hz

frequency, Hz0 2000 4000 6000100

101

102

103

fF=4000 Hz

4000Hz

p',P

a

0 2000 4000 6000100

101

102

103

Free Oscillation

500Hz

Fig. 7 a) Probe 02:02, pressure spectrum

frequency, Hz

p,P

a

0 2000 4000 6000

101

102

103

fF=1500 Hz

1500Hz

p,P

a

0 2000 4000 6000

101

102

103

Free Oscillation

1500Hz

0 2000 4000 6000

101

102

103

fF=500 Hz

500Hz

frequency, Hz0 2000 4000 6000

101

102

103

fF=4000 Hz

4000Hz

Fig 7 b) Probe 10:03, pressure spectrum

Fig. 7 c) Probe 14:10, pressure spectrum

2000 4000 600010-1

100

101 4000Hz1520HzP14:10 u, fF = 4000Hz

frequency, Hz2000 4000 600010-1

100

1014000Hz

P14:10 v, fF = 4000Hz

frequency, Hz

u,v,

w,m

/s

2000 4000 600010-1

100

1014160Hz

P14:10 v, free oscillation

u,v,

w,m

/s

2000 4000 600010-1

100

101

P14:10 u, free oscillation

Fig. 7 d) Probe 14:10, velocity spectrum

In addition to the spectral analysis presented above, the method of proper orthogonal decomposition (POD) (Lumley 1981) is employed. The POD procedure identifies the most energetic contributions and obtains the spatial structures of the corresponding modes. Here we have performed the POD analysis for the velocity field. To simplify the problem, the POD analysis is conducted only for the flowfield of an x-y cross section along the axis in the injector instead of the entire 3D flowfield. A total of 1000 snapshots extending over 10 ms are used. The

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PSDs of the time-varying coefficients of the first six most energetic modes are shown in Fig. 8. The differences of the dominant frequencies among various modes are obvious. The first four modes correspond to low-frequency flow phenomena, and the fifth and sixth modes to high-frequency flow phenomena. The spatial structures of the selected modes are shown in Fig. 9. The first and second modes represent the large structures in the flowfield. They may include vortex breakdown, separated flows from the fuel nozzle, and the modes competition between the vortex breakdown and Kelvin-Helmholtz instabilities. The third and forth modes represent the interaction/competition between the dynamic evolution of vortex breakdown and the vortex shedding due to Kelvin-Helmholtz instability (Wang, et al., 2001). The fifth and sixth modes represent the small structures corresponding to the vortex shedding arising from the Kelvin-Helmholtz instabilities in the flowfield and the frequency is around 14000Hz.

3. Unsteady Flow Evolution under Imposed External Forcing

The acoustic response of the injector to externally imposed oscillations at the inlet is studied. The imposed frequencies in the current study include 400, 500, 600, 700, 900,1500, and 4000 Hz.

The excitation is implemented by varying the mass flow rate at the inlet as [ )2sin(10 tfmm F ]πα+=

0m (4)

where , α, and fF are the mean mass flow rate, amplitude of oscillation, and forcing frequency, respectively. α is fixed at 10% in the present work.

Figure 3 shows the mean axial velocity contours for fF = 1500Hz. No discernible difference is observed between the situation with and without external forcing. The effects of forced oscillation on mean flow properties are minor in the current study. Two reasons may contribute to this phenomenon. First, the flow structures in these regions are very complex and their strengths are very high, therefore a weak forced oscillation may not affect the mean flow patterns. We can’t expect a single frequency to have a noticeable effect on the mean flowfield unless that frequency happens to cause resonance of the injector (Brereton et al, 1990). Second, the acoustic wavelengths corresponding to the forcing frequencies are much longer than the injector dimensions. Thus it is difficult to observe considerable change from the mean flowfield point of view. Figure 6 shows the contours of the mean turbulence kinetic energy at the forcing frequency of 1500Hz. The turbulence kinetic

energy in the inlet region is well organized in the presence of the forced oscillation. The strength of the separated flows near the end plate of the fuel nozzle changes and this suggests that forced oscillation may adjust the dynamic flow patterns.

Mode 01

Mode 02

Mode 04

Mode 03

Mode 05

Mode 06

4000 Hz

14000 Hz

1500 Hz

frequency, Hz

PSD

0 5000 10000 15000 20000

Fig. 8 PSD of time-varying coefficience of POD

modes (HSN)

Mode 03

-400 -320 -240 -160 -80 0 80 160 240 320 400ut: m/s

Mode 01 Mode 02

Mode 04

Mode 06Mode 05

Fig. 9 Eigenfunctions/spatial structures of POD

modes, azimuthal veloctiy contours

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Figures 7 (a-c) show the PSD of the pressure oscillations under 500Hz, 1500Hz, and 4000Hz forced oscillations at three selected probes 02:02, 10:03, and 14:10, respectively. In comparison with the case without oscillation flow, a dominant peak corresponding to the forcing frequency, i.e., 500Hz, 1500Hz, and 4000Hz, is clearly observed. However, the forcing has minor effects on the spectrum contents away from the forcing frequency. At probe 02:02, which is located outside the two high tke regions shown in Fig. 6, the effects of forcing are relative higher in comparison with the background noise. On the other hand, at probes 10:03 and 14:10, which are located in the high tke regions, the peaks do not prevail as that at probe 02:02. The reason is that the flow condition is relatively simple at probe 02:02 comparing to those at probes 10:03 and 14:10. Figures 7 (d) shows the PSD of the axial and radial velocity components at probe 14:10 under 4000Hz forced oscillation. The forcing frequency is even difficult to be identified. In other words, the effects of external excitation on the breakdown flow region are quite small.

One point needs to be noted. The pressure field is more sensitive to external forcing than the velocity field. The forcing frequency always prevails in the pressure spectrum, but it disappears in the velocity spectrum at some probe locations, for example fF = 4000Hz at probe 14:10. Actually, the velocity response is very weak for probes in the recirculation zone and the strong shear layer zone near the guide vane between the second and third passages, especially for the radial and azimuthal velocity components. The pressure oscillation propagates at the local acoustic wave speed. While the flow travels to the downstream, the azimuthal velocity oscillation decreases due to viscous and turbulent effects.

The admittance function is introduced to study the acoustic response of the exit of injector. The admittance function, A, is defined as

PpaufAγ/ˆ/ˆ

)( = (5)

where and represents the fluctuating flow speed and pressure. Figure 10 shows the radial distribution of the admittance function at the injector exit for four different forcing frequencies. The magnitude of the admittance function near the upper boundary is higher than that in the central region. The phenomenon may be attributed with the large pressure oscillation in center recirculation zone. Another important observation is the admittance function magnitude achieves its maximum at 500 Hz forcing, especially near the upper boundary. This

indicates a small pressure oscillation at 500 Hz may result in a high velocity disturbance and the velocity oscillation further affects the spray breakup in this region.

u p

The transfer function is also obtained to study the acoustic properties, which is defined as follows.

inletm

mfT|ˆ

ˆ)( =

m

(6)

where represents the fluctuating. The result is shown in Fig. 11 . The mass flux oscillation distributions are frequency dependent. For the 500 Hz forcing frequency, the distribution is almost uniform, around 1.6-2.2. For the 900 Hz case, the magnitude achieves its maximum near the boundary of the recirculation zone. At the forcing frequency of 1500Hz, the magnitude reaches 4 near the upper boundary and falls back to the range of 2 near the recirculation zone. This phenomenon indicates that the 1500Hz forcing enhances the oscillation near the upper boundary. In comparison with the case of without external forcing calculation, we also find the dominant frequency near the upper boundary is 1500Hz. On the other hand, the 4000 Hz forcing frequency doesn’t exert any influence in this region since the amplitude is low. Although a peak occurs near the boundary of the recirculation zone, where the dominant frequency at this region is around 4000Hz, for the case without external forcing, the response within the recirculation zone is very small. Based on the above observation, we find that the mass-flux fluctuation depends on the forcing frequency although the mean mass flux profiles are almost same. This redistribution process will affect

magnitude

r,m

m

0 5 10

0

5

10

15 mag.

phase

Forcing Frequency: 500 Hz

phase, radian

r,m

m

0 3.14 6.280

5

10

15Forcing Frequency: 1500 Hz

phase, radian3.14 6.28 9.42

Forcing Frequency: 4000 Hz

magnitude

0 3.14 6.28

0 5 10

Forcing Frequency: 900 Hz

Fig.10 Admittance function

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the air/fuel mixing in the downstream.

Figure 12 shows the response of the exit mass flow rate as a function forcing frequency. The inlet and outlet mass flow rates are almost in-phase at the forcing frequency of 500 Hz. As the forcing frequency increases, a phase shift is observed. Assume the streamwise length from the inlet to the outlet is L=0.03 m and the mean acoustic wave velocity is sma /400≈+u , the phase difference, θ, of the mass flow rate between the inlet and outlet satisfies the following relationship.

πθ 21Ffau

L+

≈ (7)

where fF represents the forcing frequency. The propagation of the mass flow oscillation is based on acoustic propagation. The response of the mass flow rate reaches its maximum at 1500Hz.

The radius of the CTRZ zone also varies with the forcing process. Figure 13 shows the magnitude and phase of the CTRZ radius as a function of the forcing frequency. The radius oscillation magnitude is about 10% of the mean radius, which is same as the oscillation magnitude of the mass flow rate at the injector inlet. The phase of the radius is out of phase π with that of the mass flow rate at the injector outlet. This indicates that not only the flow mass flux magnitude varies while excitation is implemented, but also the effective passage area changes. This is an interesting phenomenon and further study is required to uncover the effects of the forcing on the vortex breakdown.

mou

t/m

in

0

0.5

1

1.5

2

•

•

frequency, Hz

phas

e,ra

dian

0 1000 2000 3000 4000 5000-3.14

0

3.14

Numerical SimulationAcoustic Estimation

Fig. 12 Mass flow rate fluctuation

r,m

m

0

1

2

3

4

5

mean boundaryoscillation mag.

frequency, Hz

phas

e,ra

dian

0 1000 2000 3000 4000 5000-3.14

0

3.14

6.28

mass flow rateCTRZ boundary

phase, radian3.14 6.28 9.42

Forcing Frequency: 4000 Hz

phase, radian

r,m

m

0 3.14 6.280

5

10

15Forcing Frequency: 1500 Hz

magnitude

0 3.14 6.28

0 1 2 3 4

Forcing Frequency: 900 Hzmagnitude

r,m

m

0 1 2 3 4

0

5

10

15mag.phase

Forcing Frequency: 500 Hz

Fig. 11 Transfer function

Fig. 13 Radius fluctuation of CTRZ

Conclusions

A comprehensive numerical analysis has been conducted to investigate the vortical flow dynamics of a swirl-stabilized injector under conditions with and without external forcing. The formulation treats the unsteady, three-dimensional conservation equations, with turbulence closure achieved using a large eddy simulation (LES) technique. Detailed flow structures and injector dynamics are studied systematically. In addition, the response of the injector to externally impressed oscillations is examined as a function of forcing frequency. Results are characterized with an admittance function.

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Acknowledgements

This work was sponsored by the NASA Glenn Research Center under Grant NAG 3-2151. The support and encouragement of Kevin Breisacher is greatly appreciated.

References

Brereton, G.J., Reynolds, W.C., and Jayaraman, R. (1990), “Response of a turbulent boundary layer to sinusoidal free-stream unsteadiness”, Journal of Fluid Mechanics, vol. 221, pp. 131-159 Cannon, S., Smith, C., and Lovett, J., “LES Modeling of Combustion Dynamics in a Liquid-Fueled Flametube Combustor”, 36th AIAA/ ASME/ SAE/ASEE Joint Propulsion Conference and Exhibit, 16-19 July 2000, Huntsville, Alabama Erlebacher, G., Hussaini, M. Y., Speziale, C. G., and Zang, T. A. (1992), “Toward the Large Eddy Simulation of Compressible Turbulent Flows,” Journal of Fluid Mechanics, Vol. 238, pp. 155-185. Graves, C.B. (1997), “Outer Share Layer Swirl Mixer for A combustor”, US Patent 5-603-211 Guo, B., Langrish, T.A.G., and Fletcher, D.F., (2001) “Simulation of Turbulent Swirl Flow in Axisymmetric Sudden Expansion”, AIAA Journal Vol. 39, No. 1, 2001 96-102

Gupta, A.K., Lilley, D.G., and Syred, N. (1984), “Swirling flows”, Abacus Press. Lasheras J.C. and Hopfinger E.J. (2000), “Liquid jet instability and atomization in a coaxial gas stream”, ANNU REV FLUID MECH 32: 275- Lumley, J.L. (1981), “In Transition and Turbulence (ed. R. Meyer), pp. 215-242, Academic. Raileygh, J.W.S., The Theory of Sound, vol. II, Dover Publications, New York , 1945 Rai, M. M. and Chakravarthy, S. (1993), “Conservative High-Order Accurate Finite Difference Method for Curvilinear Grids,” AIAA 93-3380. Sarpkaya, T., (1995), “Turbulent Vortex Breakdown,” Physics of Fluids, Vol. 7, No. 10, 2301-2303 Spall R.E., and Ashby, B.M, “A Numerical Study of Vortex Breakdown in Turbulent Swirling Flows”, Journal of Fluid Engineering, 2000, Vol. 122 179-183 Wang, S., Hsieh, S.Y., and Yang, V. (2001), “Numerical Simulation of Gas Turbine Swirl-Stabilized Injector Dynamics,” AIAA 2001-0334

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